I was just thinking: in Quantum Mechanics, we start out with that whole collapsing business by observing the x position of a particle. I was thinking: why do we have to do that? What if we only observe a particle over a specific domain? Let's be more specific. I have an electron trapped in a tube. This tube has very, very thin walls. The potential in these walls is quite large but not infinite. Therefore, there is a probability the particle will quantum tunnel through these walls to the outside. If we let the particle go in the tunnel and wait a short time, this is the sort of wave function we would expect: (or should I say, the wave function's probability distribution)
Now, we shoot a photon down the tube between the two barriers (let's assume with 100% certainty the light particle will stay within the walls). There are two outcomes: the light particle keeps on going and doesn't hit the particle (the particle was observed to be outside the walls), or the photon bounces back containing information about the particle's position (the particle was observed to be inside the walls). Now, here's my real question: does the particle's wave function collapse if the particle is outside the boundary? If the particle was observed inside the walls, then it will obviously collapse (the photon contains the definite position of the particle). However, things are more messy outside the tube. Let's imagine the particle does collapse. How can we pluck apart that argument? Well, I didn't say the particle HAS to be let go inside the tube. For all that matters, this could have been the initial wave function:
For all practical purposes, the electron could be centered on the moon: so long as there is a probability of observing the particle in the tube, we will collapse it. Furthermore, if there are two particles with similar wave functions (some probability for either to be in the tube), then we will collapse them both. In conclusion, once this experiment is done, every particle in the universe (technically on the light-cone) will simultaneously be collapsed. Doesn't sound very physical, does it? Just to determine whether the 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001% chance that the particle might be observed in the tube isn't a very good reason to collapse every wave function in the universe. So, that can't be the case. Then, there is another alternative: we don't collapse the wave function (assuming the particle was not found inside the tube). How can that be the case either? If we didn't collapse the wave function in the first place, how could we know whether the particle was inside the tube or outside it? This is essentially what the problem boils down to: we must collapse the wave function in order to tell whether we need to collapse the wave function. What's the solution here? It looks hopeless either way! Thanks in advance.
P.S. There is an easy test to see what's happening, were anyone actually bothered enough to try. Let's assume the photon went right through the tube and showed the particle to be outside. Then, measure the momentum. Do this again and again. If we get a wide range of answers, we know the particle got collapsed each time. Since the particle must have a well defined position, we can't know anything about the momentum. However, if the momentum is well localized (around zero), we know the particle did not collapse. Of course, how localized "well localized" is depends on the time between measuring position and measuring momentum. Thanks again!